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\author{五六七}
\title{常微分方程第8周作业（7.2）}
%\date{2025年10月28日}

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\begin{document}

\maketitle

%\abstract{第7章第2节：例1、例子2、例子3、习题1(1)、2(1)、3。}

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\section{课堂练习(10.30)}
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\begin{enumerate}
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\item  %课堂练习1
求解一阶线性方程在原点附近的幂级数解：
\[
y' + y + x = 0.
\]

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\item  %课堂练习2
用幂级数方法求解微分方程：
\[
y'' + 4y = 0, \,\, x_0 = 0.
\]

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\item  %课堂练习3
求出下列微分方程初值问题的幂级数解的前四项：
\[
y'' + y' - 6y = 0, \,\, y(0) = 0,\,\, y'(0) = 1.
\]

\end{enumerate}

\section{课后作业}
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\begin{enumerate}
\item  %例子7.2\#1
例子7.2.1. 用幂级数方法求解 Airy 方程 
\(
y''=xy, \,\, -\infty<x<\infty. 
\)

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\item  %例子7.2\#2
例子7.2.2. 求 Airy 方程 $y''=xy$ 在 $x=1$ 附近的幂级数解。

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\item  %例子7.2\#3
例子7.2.3. 求勒让德方程在原点附近的幂级数解：
\[
(1-x^2)y''-2xy'+n(n+1)y=0.
\]

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\item  习题7.2.1(1). 
求出微分方程在 $x_0$ 附近的两个线性无关的幂级数解，
\[
y'' -xy' - y = 0, \,\, x_0=0.
\]

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\item  习题7.2.2(1). 
求出微分方程初值问题的幂级数解的前四项，
\[
y'' + xy' + y = 0, \,\, y(0)=1, y'(0)=0.
\]

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\item  习题7.2.3. 
求解埃尔米特方程 
\[
y'' -2xy' +\lambda y = 0,\,\, -\infty<x<+\infty. 
\]


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\item  引理7.1.2.  
求解初值问题
\[
\frac{dy}{dx} = \frac{1}{(1-x)(1-y)},\,\, y(0)=0.
\] 
证明该初值问题在区间 $|x|<\rho$ 内存在一个解析解，其中 
\[
\rho=1-\frac{1}{\sqrt{e}}.
\]

\end{enumerate}

\end{document}